TESLA COIL DESIGN FOR ELECTRON GUN APPLICATION

M. Paralievξ, C. Gough, S. Ivkovic Paul Scherrer Institute, Accelerator Division

5232 Villigen PSI, Switzerland

ξ email: martin.paraliev@psi.ch

Abstract The current project is to build an electron gun for X-ray Free Electron Laser (XFEL) application. The electron gun will utilize field emission and extreme accelerating gradient to achieve very low emittance. However for long-term study of cathode characteristics, a stable pulsed voltage in the megavolt range is needed. The first project phase is to design and test a 500kV pulser using a resonant air-core transformer (Tesla coil). Detailed results of simulations with Microwave Studio® and PSpice® for various coil geometries, tuning and coupling factors are given, and the optimum values for this application are given. In addition, experimental results are given for the most promising geometries.

I. INTRODUCTION The requirement to sustain very high anode-cathode gradient (0.2…1GV/m) spoke in favor of having the shortest possible high voltage pulse. The cathode field emission current I is given by Fowler-Nordheim law shown below:

−=

FcFAcI

2/3

22

1 exp ϕ, (1)

where A is emitting area, F is local field strength, ϕ work function of the emitting material, 1c and 2c are constants. Equation (1) shows that the emitted current is strongly dependant on the extracting field. In order to compare different field emitters, the pulser should have good stability and repeatability. The resonant air-core transformer technology was chosen because it is fast ( 200≈ ns), stable, fully linear and scalable. In the frequency domain, the resonant frequencies of the primary and secondary LC circuits separate with increased coupling. There is a particular coupling factor, namely 0.6, when one resonant frequency is exactly twice the other. This case is unique with its asymmetric voltage waveform and with its capability to reach the maximum voltage within one cycle (Figure 3.). This particular case of the resonant transformer we will call critically coupled. In the rest of this paper, we

describe our results of optimizing the parameters of the resonant air-core transformer.

II. RESONANT TRANSFORMER The electrical circuit of a lossless resonant pulse transformer is shown in Figure 1., where Sw is a switch, L and sL are primary and secondary inductances, C and sC are primary and secondary capacitances, and K is coupling factor. To derive the differential equations describing the electrical behavior of the circuit, the equivalent circuit shown in Figure 2. is used.

Figure 1. Simplified circuit of a resonant pulse

transformer

Figure 2. Equivalent circuit of a resonant pulse

transformer. All values are referred to the primary side.

where 1i to 5i are the currents in branches with the shown directions, 1u and 2u are respectively the voltages in primary and secondary side and 3u is the voltage across

3L , which represents the coupling. All values are referred to the primary side. The values of the components in the equivalent circuit are the following:

CCC == 21 , (2)

( )121 +== KLLL , (3)

KKLL

2

31−= . (4)

Applying Kirchoff’s rule for junctions A and B we can write:

431 iii += (5)

325 iii += (6)

Substituting the currents and making some equivalent mathematical transformations, the following system of two second order differential equations can be obtained:

022111 =−+ ukuku (7)

012212 =−+ ukuku (8)

where

)1(1

21 KLCk

−= , (9)

)1( 22 KLCKk−

= . (10)

By adding and subtracting equations (7) and (8), and using substitutions (11) and (12), the equations in the system can be decoupled [1].

121 zuu =+ (11)

221 zuu =− (12)

As a result, two independent homogeneous second order differential equations are obtained:

0)( 1211 =−+ zkkz , (13)

0)( 2212 =++ zkkz . (14)

The solutions of (13) and (14) are shown below:

( )111 cos ϕω += tAz , (15)

( )222 cos ϕω += tBz , (16)

where A, B, 1ϕ and 2ϕ are integration constants and

( )KLC +=

11

1ω , (17)

( )KLC −=

11

2ω . (18)

Using the initial conditions the final solutions for 1z and 2z can be found:

( )tUz 11 cos ω= , (19)

( )tUz 22 cos ω= , (20)

where U is the initial voltage of primary capacitor C. Going back to the original variables we find the solutions for primary voltage 1u and secondary one 2u

( ) ( )[ ]ttUu 211 coscos2

ωω += , (21)

( ) ( )[ ]ttUu 212 coscos2

ωω −= . (22)

The solution consists of two cosine functions. In order to have an asymmetric output signal (Figure 3.) we are interested in the particular case when the second frequency 2ω is exactly twice the first one 1ω . Using equations (17) and (18) the critical coupling cK can be calculated:

212 ωω = (23)

( ) ( )cc KLCKLC −=

+ 11

112 (24)

6.0=cK (25)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

0 1 2 3

Cycles

Nor

mal

ized

Am

plitu

deu1u2

Figure 3. Normalized primary (u1)

and secondary (u2) voltages

III. PARAMETRIC STUDY

In order to be able to evaluate the behavior of the transformer, two criteria have been chosen to be maximized: output peak voltage and negative to positive amplitude ratio. The equivalent circuit was simulated with PSpice®. All values are normalized to the critical coupling case ( 6.0=K ), and primary and secondary sides tuned to one and the same frequency. Two cases were covered to check the influence of damping on the circuit behavior. For the first case the damping factor was small (<5% per cycle). For the second one, the same two criteria were used but with a stronger damping. For practical reasons, a resistive loss was introduced in the primary side only. An additional restriction for the consequent pulse was used – the second oscillation peak should be 30% lower voltage than the first. In other words, the damping resistor is changed until the second peak has 70% of the amplitude of the first one (if possible). When it is not possible to meet this requirement

the chart goes to zero. Figure 4. represents the peak value of the output oscillation and Figure 5. shows negative to positive peak voltage ratio, both as function of primary capacitance and coupling factor.

Figure 4. Normalized secondary peak voltage amplitude as function of coupling and primary

capacitance with 30% loss factor

Figure 5. Negative to positive peak voltage ratio of

secondary voltage as function of coupling and primary capacitance with 30% loss factor

The graphs show that the negative to positive peak voltage ratio is more sensitive to the parameters change than the peak amplitude. Table 1. summarizes the parametric study results.

Table 1. Summarized results of the parametric study Damping <5% 30%

Absolute max. norm. voltage, % Conditions:

- Coupling - Norm. capacitor, %

120

0.60 207

115

0.60 250

Acceptable change of

coupling factor )1 , % 42

+− 7

2+−

Acceptable change of norm.

capacitance )1 , % 3434

+− 40

17+−

)1 With respect to absolute max. voltage point. Conditions: amplitude change -20%, negative to positive ratio ±10%.

IV. COIL GEOMETRY In order to find the optimal transformer layout, different transformer geometries and conductor cross sections were compared. The study was conducted using 3D PC based electromagnetic solver (Microwave Studio®) results and prototype measurements. Unfortunately the simulations appeared to be very sensitive to mesh density. One way around this problem is to keep exactly the same meshing in all the simulations independent of studied parameter. The large ratio between the dimensions of the modeled coil and conductor cross-section details results in a large number of mesh points and respectively long computing time. The results for two transformer geometries are presented: helical and spiral. Both types were measured in autotransformer mode with one turn primary. For the spiral type, central and peripheral excitation was used, as it is shown in Figure 6.

Figure 6. Central and Peripheral Excitation of spiral type transformer

V. SIMULATION AND EXPERIMENTAL

RESULTS Microwave Studio® model of 20 turns spiral coil, with outer diameter 370mm, span 3.6mm, made out of strip conductor 5mm wide and 0.9mm thick, is shown in Figure 7.

Figure 7. Microwave Studio® model

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0 5 10 15 20 25

Turns

Cou

plin

g

Measurem entSim ulation

Figure 8. Simulation vs. measurement

Graphic representation of the results based on Microwave Studio® simulations and prototype measurements are shown in Figure 8. Three parameters that affect the coupling were studied: transformer geometry, conductor cross-section profile and conductor size. A set of transformer prototypes with different geometries was built and measured (Figure 9.).

Figure 9. Measured prototypes Three transformer types were compared – helix, spiral with central excitation and spiral with peripheral excitation. To make a fair comparison the diameter of the helix was equal to the mean spiral diameter and all the other parameters were kept the same. The results are shown in Figure 10.

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10 12 14

Turns

Cou

plin

g

Spiral Peripheral

Helix

Spiral Central

Figure 10. Air-transformer geometry comparison.

To evaluate the influence of the conductor cross-section profile on the coupling, two cases with same perimeter were chosen – round and strip conductor. The results for helix and spiral transformers are shown in Figure 11.

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10 12 14

Turns

Cou

plin

g

Spiral Round Ø9Spiral Strip 14x1

Helix Round Ø9Helix Strip 14x1

Figure 11. Air-transformer conductor cross-section

profile comparison.

To evaluate the influence of the conductor size on the coupling, two strips with different width were chosen. The results for helix and spiral transformers are shown in Figure 12.

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10 12 14Turns

Cou

plin

g

Spiral Strip 50x1Spiral Strip 14x1Helix Strip 50x1Helix Strip 14x1

Figure 12. Air-transformer conductor

size comparison.

VI. CONCLUSIONS

The performed parametric study showed that the output pulse amplitude and shape are sensitive to the coupling and relatively less sensitive to primary capacitance. To keep large negative to positive peak voltage ratio of the output pulse, the coupling factor should be kept within 0.58 to 0.62. Slightly larger value of primary capacitance (with respect to the tuned value) gives an improvement in amplitude without changing the negative to positive ratio too much. With increasing number of turns the coupling decreases. The spiral transformer with peripheral excitation gives the best coupling for given dimensions and conductors span. The influence of the conductor cross-section profile on coupling is small on condition that the conductor cross-section perimeter is the same. In general wider strips give better coupling. The difference in coupling between simulations (with fixed mesh) and measurements is less than 1%. In general, scaling the size of the transformer gives a linear scaling of all self and mutual inductances but does not change the coupling.

VII. REFERENCES [1] Frank L. H. Wolfs, Physics Lecture Notes 2004, University of Rochester, Rochester, NY 14627, USA